解：a^4=(a²)²=(a+1)²=a²+2a+1=(a+1)+2a+1=3a+2a^5=a(3a+2)=3a²+2a=3(a+1)+2a=5a+3.同理：b^5=5b+3，又a、b是一元二次方程X²-X-1=0的两根，∴a+b=1，∴a^5+b^5=5(a+b)+6=11.
